Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Opening

15 minutes

As Day 2 of this lesson begins, students share out their ideas of situations for their linear programming problems. I begin class by asking students why the word linear is used in the phrase linear programming. We then discuss what is “linear” about linear programming and check their problems for linearity. Students should be able to identify that the constraints and the expression being maximized or minimized are both linear. I explain to students that it will be important in the problems they are writing that their constraints and whatever they are maximizing or minimizing are linear.

Once the seed of linearity is firmly planted, the students examine each other’s work to make sure the constraints are linear conditions of the variables they have chosen and that whatever they are maximizing or minimizing depends linearly on the variables as well. They then begin to solve the problem they created.

Next, I ask each group each group to share their ideas for a problem situation. I make a list of what they have chosen for variables, constraints, and what thing they will maximize or minimize. I ask students to check and make sure that the thing each group is maximizing or minimizing depends linearly on the variables they chose. They should also check to make sure the constraints are linear conditions of the variables as well. I ask for volunteers to help adjust the constraints or max/min things to make them linear if they are not already.

Investigation

Students should now work in the same groups they work in on Day 1 of the lesson. Each group should be ready to either write out their word problem or start by writing inequalities for their constraints to see the feasible region. I let groups decide what order they want to work in. Many groups may realize they have to change some aspects of their problem. I try to let them come to this realization on their own, rather than pointing it out to them earlier in the process.

Some groups will have extraneous constraints. I ask them how they could change that particular constraint to allow it to have an impact on the feasible region.

Groups that are looking to minimize something that have a feasible region right around the origin may need to change their constraints or switch to maximizing something.

Some groups may come up with constraints that only use one of their variables. This may confuse them when they are graphing. I remind them they have seen inequalities like this before. Also, I encourage them to think of this constraint by itself, without considering the other variable.

Writing Linear Programming Problems.doc

Writing Linear Programming Problems.pdf

Closing + Homework

5 minutes

I explain to students that tomorrow they will begin to prepare their presentations and write-up of the problem they created.

Reflection/Exit ticket: I ask students to write down what they contributed to their group today.

Homework: Students should have a graph of their feasible region by the end of class. If they haven’t finished yet, or are still adjusting their problem, I assign this for homework. Some students may have their inequalities and a graph, but have not written out the word problem yet. This is also an appropriate homework assignment. To hear more about how I try to have all students on the same page for the next lesson, check out this video clip: Writing Linear Programming Problems !